Question

The volume of a sphere is $\dfrac{4}{3}\pi {r^3}$ cubic units then the ratio of the volume of a cube to that of a sphere which will fit inside the cube is
A. $\dfrac{4}{3}:\pi $
B. $6:\pi $
C. $4:3$
D. $4:\pi $

Answer 1

Hint: It can be easily understood that in order for the sphere to fit inside the cube, diameter of the sphere must be equal to the side of the cube. Use the fact that radius is half of the diameter radius of the sphere can be found. Then all you need is to find the volume of the sphere and the cube and take the ratio of the volume of the cube to the volume of the sphere to get the desired ratio.


Complete step-by-step solution
Let us consider the volume of the sphere, which s $\dfrac{4}{3}\pi {r^3}$ cubic units.

Our objective is to determine the ratio of the volume of a cube to that of a sphere which will fit inside the cube. For that, let us take the side of the cube to be ‘a’. As we know that all the sides of the cube are equal, thus all the sides of the cube are equal to ‘a’.

We know that the volume of the cube is given by the formula, $V = {s^3}$, where ‘s’ is the side of the cube.
$ \Rightarrow {V_c} = {a^3}$
Where ${V_c}$ denotes the volume of the cube.


In order for the sphere to fit inside the cube, its diameter should be equal to the side of the cube, which is ‘a’. And we know that the radius of the sphere is half of the diameter.
$
   \Rightarrow r = \dfrac{d}{2} \\
   \Rightarrow r = \dfrac{a}{2} \\
$

Now, we know that the volume of the sphere of radius ‘r’ is $\dfrac{4}{3}\pi {r^3}$. Thus for the obtained radius the required volume will be obtained by replacing ‘r’ by $\dfrac{a}{2}$ in the formula for the volume of the sphere.
$
   \Rightarrow {V_s} = \dfrac{4}{3}\pi {\left( {\dfrac{a}{2}} \right)^3} \\
   \Rightarrow {V_s} = \dfrac{4}{3}\pi \times \dfrac{{{a^3}}}{8} \\
   \Rightarrow {V_s} = \dfrac{{\pi {a^3}}}{{3 \times 2}} \\
   \Rightarrow {V_s} = \dfrac{{\pi {a^3}}}{6} \\
$
Where ${V_s}$ denotes the volume of the sphere.

Finally we take the ratio of the volume of the cube given by ${V_c}$ and the volume of the sphere given by ${V_s}$, to get the final answer.
$
  \dfrac{{{V_c}}}{{{V_s}}} = \dfrac{{{a^3}}}{{\dfrac{{\pi {a^3}}}{6}}} \\
   \Rightarrow \dfrac{{{V_c}}}{{{V_s}}} = \dfrac{6}{\pi } \\
$

Hence, the ratio of the volume of a cube to that of a sphere which will fit inside the cube is $6:\pi $, which is option (B).

Note: While finding the volumes, the correct formula should be applied, do not mix up the formula in the conic. You can star mark the fact that in order for the sphere to fit in the cube, it should have the diameter equal to that of the side of the cube, neither more nor less. Also, while taking the ratio, carefully do the volume of the cube to the volume of the sphere and not the other way around.